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2017 East Central NA Regional Contest

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2017-10-28 19:00 UTC

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Problem BCraters

General Warren Pierce has a bit of a problem. He’s in charge
of a new type of drone-delivered explosive and they’ve been
testing it out in the Nevada desert, far enough from any
population center to avoid civilian casualties and prying eyes.
Unfortunately word has gotten out about these experiments and
now there’s the possibility of careless on-lookers, nefarious
spies, or even worse — nosy reporters! To keep them away from
the testing area, Warren wants to erect a single fence
surrounding all of the circular craters produced by the
explosions. However, due to various funding cuts (to support
tax cuts for the you-know-who) he can’t just put up miles and
miles of fencing like in the good old days. He figures that if
he can keep people at least $10$ yards away from any crater he’ll
be okay, but he’s unsure of how much fencing to request. Given
the locations and sizes of the craters, can you help the
General determine the minimum amount of fencing he needs? An
example with three craters (specified in Sample Input 1) is
shown below.

Figure 1: Three craters with a fence around them.

Input

The first line of input contains a single positive integer
$n$$(n \leq 200)$, the number of craters.
After this are $n$ lines
specifying the location and radius of each crater. Each of
these lines contains 3 integers $x$$y$$r$, where $x$ and $y$ specify the location of a crater
$(|x,y| \leq 10\, 000)$
and $r$ is its radius
($0 < r \leq 5\, 000$).
All units are in yards.

Output

Display the minimum amount of fencing (in yards) needed to
cordon off the craters, with an absolute or relative error of
at most $10^{-6}$.